quiz9soln - MATH 231 Spring 2008 Name Key 1(4 points...

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MATH 231 / Spring 2008 April 2, 2008 Name: Key 1. (4 points) Convert the point ( x, y, z ) = parenleftBig 1 2 , 1 2 , 1 2 parenrightBig into cylindrical and spherical. Solution: For cylindrical, we’re already part of the way there: we know z . Thus, we only need compute r and θ . We have an equation for r ; namely, r 2 = ( 1 2 ) 2 + ( 1 2 ) 2 = 1 2 , so r = 1 2 . Furthermore, we know that θ = tan - 1 (1) = π/ 4. Now for spherical. For spherical, we know that ρ 2 = x 2 + y 2 + z 2 = r 2 + z 2 = 1 2 + 1 2 = 1. Also, we already computed θ in our cylindrical, so we only need to compute φ . We know that φ = tan - 1 ( r z ) = tan - 1 (1) = π/ 4. 2. (6 points) Express the integral integraltextintegraltextintegraltext E f ( x, y, z ) dV as an iterated integral in 6 different ways, where E is the solid bounded by the surfaces x = 2, y = 2, z = 0, and x + y - 2 z = 2. Solution: First, let’s graph the region that we’re looking at: 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 Note that I have set this up in a rather non-standard orientation for better viewing.
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